On the Σ-invariants of Artin groups

The BNS-invariant for some Artin groups of arbitrary circuit rank

“…Finally, the next result is an easy consequence of the application of Bieri-Renz's theorem on Artin groups. 2 Proposition A implies Theorem B Note that by [12] if G is an Artin group with underlying graph G and µ :…”

“…Note that by [12] if G is an Artin group with underlying graph G and µ : G → R a non-zero real character of G such that L(µ) is a connected dominant subgraph of G then [µ] ∈ Σ 1 (G). Observe that under the assumptions of Theorem B if µ : G → R is a non-zero homomorphism then µ(u 1 ) = µ(u 2 ) = · · · = µ(u n ).…”

“…Recall that as stated above by [12] if L(µ) is a connected dominant subgraph of G then [µ] ∈ Σ 1 (G). By the last two paragraphs if L(µ) is not a connected dominant subgraph of G then either µ = χ or µ = −χ, where χ is the character considered in Proposition A.…”

On graphic arrangement groups

“…The groups P X , X ∈ Λ Γ , are isomorphic to F 2 × Z, yielding the first statement. The second follows from [24], with the observation that the clique complexes of the graphs associated to the right-angled Artin groups (F 2 ) m and (F 2 ) m × Z m have the same connectivity.…”

“…Now suppose G S is a free group for every S ∈ X . Then one can apply the results of [24] to identify the homological finiteness type of ρ X (G). Refer to [6] for the details of the computation.…”